In this course we will cover the fundamentals of the theory of von Neumann algebras, including von Neumann's bicommutant theorem and Kaplansky's density theorem, comparison of projections and the type classification as well as modular theory. If time permits, we will also have a brief look at finite-dimensional approximations and amenability of von Neumann algebras. More details to follow.
This course can be taken as module 10-MAT-MM3FA in the M. Sc. Mathematics, 10-MAT-MPFOP2 in the M. Sc. Mathematical physics or as "Hauptvorlesung Funktionalanalysis" in the diploma Mathematics program.
Literature:
The lecture will be accompanied by a block seminar at the end of the semester on Tsirelson's problem. Tsirelson's problem was, until its recent resolution, one of the most important open questions in quantum information theory. Roughly speaking, it asks whether the tensor product model of quantum mechanics and the commuting operator model of algebraic quantum field theory give rise to the same correlations for spatially separated observables. In recent decades, it has transpired that Tsirelson's problem has deep connections to operator algebra theory and is in fact equivalent to one of the Connes–Kirchberg problem, one of the deep open problems in C*-algebra and von Neumann algebra theory.
The exact date of the seminar will be decided in the first weeks of the semester.
This is the introductory course for the master's program Mathematical Physics, taught jointly with Prof. Gajic (ITP). The mathematics part will cover a bit of everything, ranging from metric spaces and multivariate analysis (Banach fixed-point theorem, inverse and implicit function theorem) over manifolds and integration of differential forms (Stokes's theorem) and measure and integration theory (convergence theorems for the Lebesgue integral) to operator theory on Hilbert space and spectral theory (spectral theorem for unbounded self-adjoint operators).
Literature:
The two main topics of this course are abstract spectral theory and Schatten classes. In the first part, we will study commutative C*-algebras, culminating in the Gelfand representation theorem. In the second part, our focus lies on certain classes of bounded operators on Hilbert space that are related to the trace, namely trace-class operators, Hilbert-Schmidt operators etc. All these operators are compact, so we will also look into this broader class a bit. Additionally, I also plan to include some fundamentals of functional analysis that were not covered in Mathematical Physis 2, namely the Hahn-Banach theorem and the Stone-Weierstraß theorem.
In the seminar, we will look at properties of entropy in quantum information. Topics for talks include quantum states and quantum channels, quantum Stein lemma and Uhlmann's monotonicity theorem (also known as data processing inequality).
Required background is a solid understanding of functional analysis as covered in the course Mathematical Physics 2. If you did not take this course with me, have a look at Chapter 4 of the Lecture Notes posted below.
Notes: Lecture Notes
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Time:
Thursday 11:00 - 13:00 (SG 312), Friday 9:00 - 11:00 (SG214)
This is the introductory course for the master's program Mathematical Physics, taught jointly with Prof. Gajic (ITP). The mathematics part will cover a bit of everything, ranging from metric spaces and multivariate analysis (Banach fixed-point theorem, inverse and implicit function theorem) over manifolds and integration of differential forms (Stokes's theorem) and measure and integration theory (convergence theorems for the Lebesgue integral) to operator theory on Hilbert space and spectral theory (spectral theorem for unbounded self-adjoint operators).
This course is administered via moodle.
Literature:
Time:
Lectures: Monday, Tuesday 11:00 - 13:00, Thursday, Friday 9:00 - 11:00
Recitations: Tuesday 15:00 - 17:00, Thursday 11:00 - 13:00
Course at University of Paderborn
Course at ISTA
Course at ISTA, jointly taught with Haonan Zhang